A computational approach for solving 𝑦²=1^{𝑘}+2^{𝑘}+…+𝑥^{𝑘}
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Bibliographic record
Abstract
We present a computational approach for finding all integral solutions of the equation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="y squared equals 1 Superscript k Baseline plus 2 Superscript k Baseline plus midline-horizontal-ellipsis plus x Superscript k"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>y</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>=</mml:mo> <mml:msup> <mml:mn>1</mml:mn> <mml:mi>k</mml:mi> </mml:msup> <mml:mo>+</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>k</mml:mi> </mml:msup> <mml:mo>+</mml:mo> <mml:mo> ⋯ </mml:mo> <mml:mo>+</mml:mo> <mml:msup> <mml:mi>x</mml:mi> <mml:mi>k</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">y^2=1^k+2^k+\dotsb +x^k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for even values of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . By reducing this problem to that of finding integral solutions of a certain class of quartic equations closely related to the Pell equations, we are able to apply the powerful computational machinery related to quadratic number fields. Using our approach, we determine all integral solutions for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 less-than-or-equal-to k less-than-or-equal-to 70"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo> ≤ </mml:mo> <mml:mi>k</mml:mi> <mml:mo> ≤ </mml:mo> <mml:mn>70</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">2\le k\le 70</mml:annotation> </mml:semantics> </mml:math> </inline-formula> assuming the Generalized Riemann Hypothesis, and for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 less-than-or-equal-to k less-than-or-equal-to 58"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo> ≤ </mml:mo> <mml:mi>k</mml:mi> <mml:mo> ≤ </mml:mo> <mml:mn>58</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">2\le k\le 58</mml:annotation> </mml:semantics> </mml:math> </inline-formula> unconditionally.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it